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Global attractivity and uniform persistence in Nicholson’s blowflies. (English) Zbl 0869.34056
Summary: The delay differential equation \[ \dot N(t)= -\delta N(t)+ PN(t-\tau)e^{-aN(t-\tau)},\quad t\geq 0\tag{\(*\)} \] was used by W. S. C. Gurney, S. P. Blythe and R. M. Nisbet [Nature 287, 17-21 (1980)] in describing the dynamics of Nicholson’s blowflies. We show that for \(P\leq\delta\), every nonnegative solution of \((*)\) tends to zero as \(t\to\infty\). On the other hand, for \(P>\delta\), \((*)\) is uniformly persistent. Moreover, if in addition, \((e^{\delta r}- 1)\ln{P\over\delta}<1\), then every positive solution of \((*)\) tends to \(N^*={1\over a}\ln {P\over \delta}\) as \(t\to\infty\).

34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
92D25 Population dynamics (general)
92D40 Ecology